GRAPH DECOMPOSITIONS WITHOUT ISOLATED VERTICES .3.

Let G be a graph of order n, and n = Sigma(i = 1)(k) a(i) be a partiti on of n with a(i) greater than or equal to 2. In this article we show that if the minimum degree of G is at least 3k - 2, then for any disti nct k vortices v(1),...,v(k) of G, the vertex set V(G) can be decompos ed into k disjoint subsets A(1),...,A(k) so that \A(i)\ = a(i),v(i) is an element of A(i) and ''the subgraph induced by A(i) contains no iso lated vertices'' for all i, 1 less than or equal to i less than or equ al to k. Here, the bound on the minimum degree is sharp. (C) 1997 John Wiley & Sons, Inc.